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Skedastic Function
In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function. Conditional variances are important parts of autoregressive conditional heteroskedasticity (ARCH) models. Definition The conditional variance of a random variable ''Y'' given another random variable ''X'' is :\operatorname(Y\mid X) = \operatorname\Big(\big(Y - \operatorname(Y\mid X)\big)^\;\Big, \; X\Big). The conditional variance tells us how much variance is left if we use \operatorname(Y\mid X) to "predict" ''Y''. Here, as usual, \operatorname(Y\mid X) stands for the conditional expectation of ''Y'' given ''X'', which we may recall, is a random variable itself (a function of ''X'', determined up to probability one). As a result, \operatorname(Y\mid X) itself is a random variable (and is a function of ''X''). Expla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Deviation And Dispersion
Statistics (from German: ', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Effects Model
In econometrics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. A random effects model is a special case of a mixed model. Contrast this to the biostatistics definitions, as biostatisticians use "fixed" and "random" effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables). Qualitative description Random effect models assist in controlling for unobserved heterogeneity when the heterogeneity is constant over time and not correlated with independent variables. This constant can be removed from longitudinal data through differencing, since taking a first difference will remove any time invariant components of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Model
A mixed model, mixed-effects model or mixed error-component model is a statistical model containing both fixed effects and random effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences. They are particularly useful in settings where repeated measures design, repeated measurements are made on the same statistical units (see also longitudinal study), or where measurements are made on clusters of related statistical units. Mixed models are often preferred over traditional analysis of variance regression models because they don't rely on the independent observations assumption. Further, they have their flexibility in dealing with missing values and uneven spacing of repeated measurements. The Mixed model analysis allows measurements to be explicitly modeled in a wider variety of correlation and variance-covariance variance, avoiding biased estimations structures. This page will discuss mainly linear mixed-effects models rathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Total Variance
The law of total variance is a fundamental result in probability theory that expresses the variance of a random variable in terms of its conditional variances and conditional means given another random variable . Informally, it states that the overall variability of can be split into an “unexplained” component (the average of within-group variances) and an “explained” component (the variance of group means). Formally, if and are random variables on the same probability space, and has finite variance, then: \operatorname(Y) \;=\; \operatorname\bigl operatorname(Y \mid X)\bigr\;+\; \operatorname\!\bigl(\operatorname \mid Xbigr).\! This identity is also known as the variance decomposition formula, the conditional variance formula, the law of iterated variances, or colloquially as Eve’s law, in parallel to the “Adam’s law” naming for the law of total expectation. In actuarial science (particularly in credibility theory), the two terms \operatorname operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Density
In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value x of X as a parameter. When both X and Y are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable. If the conditional distribution of Y given X is a continuous distribution, then its probability density function is known as the conditional density function. The proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Distribution
Conditional (if then) may refer to: * Causal conditional, if X then Y, where X is a cause of Y *Conditional probability, the probability of an event A given that another event B * Conditional proof, in logic: a proof that asserts a conditional, and proves that the antecedent leads to the consequent *Material conditional, in propositional calculus, or logical calculus in mathematics * Relevance conditional, in relevance logic * Conditional (computer programming), a statement or expression in computer programming languages *A conditional expression in computer programming languages such as ?: *Conditions in a contract A contract is an agreement that specifies certain legally enforceable rights and obligations pertaining to two or more parties. A contract typically involves consent to transfer of goods, services, money, or promise to transfer any of thos ... Grammar and linguistics * Conditional mood (or conditional tense), a verb form in many languages * Conditional se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of a set, partition of this probability space. Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted E(X\mid Y) analogously to conditional probability. The function form is either denoted E(X\mid Y=y) or a separate function symbol such as f(y) is introduced with the meaning E(X\mid Y) = f(Y). Examples Example 1: Dice rolling Consider the roll of a fair die and let ''A'' = 1 if the number is even (i.e., 2, 4, or 6) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which * the domain is the set of possible outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Total Expectation
The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing property of conditional expectation, among other names, states that if X is a random variable whose expected value \operatorname(X) is defined, and Y is any random variable on the same probability space, then :\operatorname (X) = \operatorname ( \operatorname ( X \mid Y)), i.e., the expected value of the conditional expected value of X given Y is the same as the expected value of X. The conditional expected value \operatorname( X \mid Y ), with Y a random variable, is not a simple number; it is a random variable whose value depends on the value of Y. That is, the conditional expected value of X given the ''event'' Y = y is a number and it is a function of y. If we write g(y) for the value of \operatorname ( X \mid Y = y) then the random variable \operatorname( X \mid Y ) is g( Y ) . One special case states that if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
